How to create perpendicular edges along one direction?
You can achieve that by creating a plane and subdividing it using Ctrl + R into parallel equally spaced edges, then using Knife Project tool, you can project your object to the plane, all you need to do later is to remove all the faces that fall outside the projected faces.
But looking at your file, you need first to turn your object into a mesh using Alt+C and choose Mesh from curve, then in edit mode, select all, press X and choose Limited Disolve, in order to reduce the vertices in your object.
Here's your edited file: